Triangulate a Polygon

POLYGON_TRIANGULATE is a Python library which triangulates a possibly nonconvex polygon in 2D, and which can use gnuplot to display the external edges and internal diagonals of the triangulation.

The polygon is defined by an input file which gives the coordinates of the vertices of the polygon, in counterclockwise order.

No consecutive pair of vertices should be equal; when describing a polygon, sometimes the first and last vertices are equal. For this program, that is not the case. To describe a square, your input file should contain four pairs of coordinates, for instance.

The vertices should be listed in counterclockwise order. If you list them in clockwise order, then the function will refuse to operate on the data.

It is possible to create a polygon that has zero area. The function will refuse to process such an object.

This function cannot triangulate a polygon which includes one or more "holes". That is actually a significantly more difficult task.

The polygon does not need to be convex. However, you should be careful not to specify a polygon which crosses itself, since this means the interior of the polygon is not well defined, and hence a triangulation is not well defined. As a simple example of such a problem, consider the four vertices of a square in counterclockwise order: V1, V2, V3, V4, and list them instead as V1, V3, V2, V4. This shape cannot be triangulated in the usual sense. However, the function may not realize this, in which case it will return what it thinks is a reasonable triangulation of the (unreasonable) data.

The output of the program is a list of the N-3 triples of nodes that form the triangles of the triangulation.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


POLYGON_TRIANGULATE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

POLYGON_GRID, a Python library which generates a grid of points over the interior of a polygon in 2D.

POLYGON_INTEGRALS, a Python library which returns the exact value of the integral of any monomial over the interior of a polygon in 2D.

POLYGON_MONTE_CARLO, a Python library which applies a Monte Carlo method to estimate the integral of a function over the interior of a polygon in 2D.

POLYGON_PROPERTIES, a Python library which computes properties of an arbitrary polygon in the plane, defined by a sequence of vertices, including interior angles, area, centroid, containment of a point, convexity, diameter, distance to a point, inradius, lattice area, nearest point in set, outradius, uniform sampling.

TOMS112, a Python library which determines whether a point is contained in a polygon, by Moshe Shimrat. This is a version of ACM TOMS algorithm 112.


Based on a C function by Joseph ORourke; Python version by John Burkardt.


Source Code:

Examples and Tests:

You can go up one level to the Python source codes.

Last revised on 10 September 2016.